Properties

Label 230.4.a.e
Level $230$
Weight $4$
Character orbit 230.a
Self dual yes
Analytic conductor $13.570$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.5704393013\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{2} + q^{3} + 4 q^{4} - 5 q^{5} + 2 q^{6} - 18 q^{7} + 8 q^{8} - 26 q^{9} + O(q^{10}) \) \( q + 2 q^{2} + q^{3} + 4 q^{4} - 5 q^{5} + 2 q^{6} - 18 q^{7} + 8 q^{8} - 26 q^{9} - 10 q^{10} - 32 q^{11} + 4 q^{12} - 47 q^{13} - 36 q^{14} - 5 q^{15} + 16 q^{16} + 20 q^{17} - 52 q^{18} + 36 q^{19} - 20 q^{20} - 18 q^{21} - 64 q^{22} - 23 q^{23} + 8 q^{24} + 25 q^{25} - 94 q^{26} - 53 q^{27} - 72 q^{28} - 27 q^{29} - 10 q^{30} - 33 q^{31} + 32 q^{32} - 32 q^{33} + 40 q^{34} + 90 q^{35} - 104 q^{36} + 56 q^{37} + 72 q^{38} - 47 q^{39} - 40 q^{40} - 157 q^{41} - 36 q^{42} + 18 q^{43} - 128 q^{44} + 130 q^{45} - 46 q^{46} + 65 q^{47} + 16 q^{48} - 19 q^{49} + 50 q^{50} + 20 q^{51} - 188 q^{52} - 14 q^{53} - 106 q^{54} + 160 q^{55} - 144 q^{56} + 36 q^{57} - 54 q^{58} - 744 q^{59} - 20 q^{60} + 552 q^{61} - 66 q^{62} + 468 q^{63} + 64 q^{64} + 235 q^{65} - 64 q^{66} - 156 q^{67} + 80 q^{68} - 23 q^{69} + 180 q^{70} + 699 q^{71} - 208 q^{72} - 609 q^{73} + 112 q^{74} + 25 q^{75} + 144 q^{76} + 576 q^{77} - 94 q^{78} - 644 q^{79} - 80 q^{80} + 649 q^{81} - 314 q^{82} + 512 q^{83} - 72 q^{84} - 100 q^{85} + 36 q^{86} - 27 q^{87} - 256 q^{88} - 102 q^{89} + 260 q^{90} + 846 q^{91} - 92 q^{92} - 33 q^{93} + 130 q^{94} - 180 q^{95} + 32 q^{96} + 578 q^{97} - 38 q^{98} + 832 q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
2.00000 1.00000 4.00000 −5.00000 2.00000 −18.0000 8.00000 −26.0000 −10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.4.a.e 1
3.b odd 2 1 2070.4.a.e 1
4.b odd 2 1 1840.4.a.d 1
5.b even 2 1 1150.4.a.b 1
5.c odd 4 2 1150.4.b.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.e 1 1.a even 1 1 trivial
1150.4.a.b 1 5.b even 2 1
1150.4.b.f 2 5.c odd 4 2
1840.4.a.d 1 4.b odd 2 1
2070.4.a.e 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(230))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( -1 + T \)
$5$ \( 5 + T \)
$7$ \( 18 + T \)
$11$ \( 32 + T \)
$13$ \( 47 + T \)
$17$ \( -20 + T \)
$19$ \( -36 + T \)
$23$ \( 23 + T \)
$29$ \( 27 + T \)
$31$ \( 33 + T \)
$37$ \( -56 + T \)
$41$ \( 157 + T \)
$43$ \( -18 + T \)
$47$ \( -65 + T \)
$53$ \( 14 + T \)
$59$ \( 744 + T \)
$61$ \( -552 + T \)
$67$ \( 156 + T \)
$71$ \( -699 + T \)
$73$ \( 609 + T \)
$79$ \( 644 + T \)
$83$ \( -512 + T \)
$89$ \( 102 + T \)
$97$ \( -578 + T \)
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